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Limacon

1911 Encyclopedia Britannica

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(from the Lat. limax, a slug), a curve invented by Blaise Pascal and further investigated and named by Gilles Personne de Roberval. It is generated by the extremities of a rod which is constrained to move so that its middle point traces out a circle, the rod always passing through a fixed point on the circumference. The polar equation is r=a+b cos 0, where 2a= length of the rod, and b= diameter of the circle. The curve may be regarded as an epitrochoid (see Epicycloid) in which the rolling and fixed circles have equal radii. It is the inverse of a central conic for the focus, and the first positive pedal of a circle for any point. The form of the limacon depends on the ratio of the two constants; if a be greater than b, the curve lies entirely outside the circle; if a equals b, it is known as a cardioid; if a is less than b, the curve has a node within the circle; the particular case when b= 2a is known as the trisectrix. In the figure (1) is a limagon, (2) the cardioid, (3) the trisectrix.

Properties of the limagon may be deduced from its mechanical construction; thus the length of a focal chord is constant and the normals at the extremities of a focal chord intersect on a fixed circle. The area is (b 2 +a 2 /2)7r, and the length is expressible as an elliptic integral.